2. Method

2.1. General remarks

To compare more or less heterogeneous photometric and spectrophotometric data, obtained at different epochs by different observers and systems, we reduced them all to a common set of spectrophotometric quantities. We transformed them to the spectrophotometric parameters derived in the BCD system. These are: (a) gradient of the continuum energy distribution in the spectral domain m; (b) total BD of Be stars, D, measured at m, and (c) V magnitude as defined in the standard UBV system. As known, the total BD of Be stars has two components: D = D_* + d, where accounts for the invariable stellar BD, and d is the variable component due to the CE. It is d < 0 for Balmer continuum emission excess, and d > 0 for Balmer continuum deficiency. In this paper we study the total BD only.

2.2. The (, D, V) parameters and the structure of the CE

Reduction of all data obtained from quite different photometric systems to the BCD parametric triptych (, D, V), allows us: (a) to have an uniform set of parameters with a clear physical meaning and easy to interpret theoretically; (b) to test previous conclusions obtained with the BCD data on long-term spectrophotometric behaviour of Be stars. Interpretation of the (, D, V) parameters in terms of CE opacity, extent and temperature will be sketched in Sect. 4.

2.3. Photometric systems of compiled data

Data of Be stars used in this work were obtained in (or reduced in the original publications to) the following photometric systems: (a) the UBV standard system (Johnson & Morgan 1951, 1953); (b) the UBVRI system (Johnson 1965; Cousins 1976, 1978); (c) the seven-colour (U,B,V,B1,B2,V1,G) Geneva photometric system (Golay 1963; Rufener & Maeder 1971; Rufener & Nicolet 1988; Rufener 1979); (d) the uvby Strömgren system (Strömgren 1966; Grønbech et al. 1976; Grønbech & Olsen 1977); (e) 13-colour photometry (Johnson & Mitchell 1975); (f) photographic spectrophotometric data (BCD).

 

HD HR Sp.Type D_* N References

Name dex

o Cas 0.06 0.006 0.02 114,115,142,198

5394 264 2.41 -0.038 0.97 28070-47528 389 7,12,13,14,15,32,38,67,80,

0.22 0.054 0.13 101,104,109,110,111,112, 114,115,142,155,198

23862 1180 5.17 0.399 1.01 33976-48341 341 2,11,12,13,14,19,33,35,42,

28 Tau 0.09 0.079 0.06 53,56,65,80,85,99,100,110, 114,115,116,158,172,178,196 197,198

24534 1209 6.55 0.000 1.44 34271-49029 331 2,13,17,18,38,43,66,67,69,

X Per 0.21 0.045 0.12 87,94,95,105,106,118,127, 141,142,147,151,164,198

25940 1273 4.04 0.225 1.06 35400-40560 18 32,38,80,81,82,87,110,114,

48 Per 0.02 0.013 0.02 115,142,144,198

28497 1423 5.51 0.066 0.81 35400-48202 92 6,26,37,58,64,68,73,110,

DU Eri 0.04 0.016 0.03 142,157,198

32343 1622 5.13 0.152 0.97 32809-45697 25 38,51,110,128,142,155,196,

11 Cam 0.09 0.023 0.02 198

33328 1679 4.26 0.113 0.79 37666-49278 404 3,26,30,38,68,78,110,114,

0.02 0.006 0.02 115,139,140,159,181,182,198

35439 1789 4.80 0.065 0.86 33966-48313 117 2,8,22,30,38,73,80,110,114,

25 Ori 0.10 0.026 0.07 115,126,142,143,174,191,198

37202 1910 2.97 0.178 0.80 35400-48974 78 32,38,59,76,80,85,110,114

0.05 0.026 0.04 115,142,198

37967 1961 6.23 0.191 1.00 35400-48600 26 32,38,85,110,142,193,198

V731 Tau 0.05 0.012 0.03

48914 B5Ib-II 0.200 7.27 0.300 1.12 41329-47934 640 55,184,187,198

0.07 0.035 0.03

48917 2492 5.21 0.069 0.93 38441-49407 406 22,37,45,57,59,60,62,73,139,

10 CMa 0.06 0.015 0.04 140,179,181,182,185,198

50123 2545 5.75 0.158 1.36 40967-49401 186 73,92,125,139,140,179,181,

0.03 0.008 0.03 182,198

56014 2745 4.59 0.146 0.81 38104-47232 182 6,22,58,60,62,73,114,139,

27 CMa 0.08 0.030 0.03 140,181,182,198

56139 2749 3.77 0.164 0.92 38384-48349 182 6,22,26,45,59,60,62,78,108,

0.17 0.030 0.08 114,139,140,179,181,198

58050 2817 6.56 0.145 0.76 35400-48344 41 38,110,142,163,170,198

0.09 0.010 0.02

58978 2855 5.62 0.031 0.95 38135-49399 275 1,22,24,37,57,58,59,62,64,

FY CMa 0.04 0.006 0.02 73,74,91,131,139,140,179, 181,182,198

60848 6.85 0.009 0.85 35571-48347 14 48,74,135,198

BN Gem 0.07 0.027 0.06

60855 2921 B3IVe 0.190 5.69 0.171 0.91 41484-45746 19 26,31,37,59,73,97,134,136,

0.03 0.011 0.03 142,169,170,175,198

63462 3034 4.49 -0.000 1.25 38076-49439 180 26,30,36,58,59,73,98,114,

o Pup 0.02 0.007 0.02 140,143,179,181,182,198

65875 3135 6.51 0.127 0.94 35400-48178 23 38,58,59,73,110,132,142,

0.05 0.023 0.05 170,187,189,198

68980 3237 4.73 0.036 0.94 38076-49399 155 26,30,58,59,60,61,62,64,78,

MX Pup 0.09 0.016 0.04 139,140,179,181,182,192,198

83953 3858 4.76 0.238 0.89 35400-48972 66 8,26,30,36,58,59,64,73,78,

0.02 0.011 0.02 114,129,140,142,150,179, 181,182,198

89890 4074 B4III-IVe 0.235 4.49 0.252 0.91 37666-49400 153 26,30,36,44,64,73,139,140,

0.01 0.007 0.02 179,181,182,198

 

HD HR Sp.Type D_* N References

Name dex

0.01 0.008 0.02

91465 4140 3.31 0.166 0.90 38807-48258 23 23,27,36,58,179,181,198

PP car 0.08 0.013 0.04

109387 4787 3.87 0.243 0.88 35400-47110 201 2,10,19,20,38,42,75,79,80,

0.05 0.020 0.05 110,114,115,117,142,154, 170,198

120991 5223 B2IIIe 0.140 6.07 0.063 0.98 38472-48851 229 23,37,58,62,63,64,72,73,

V767 Cen 0.14 0.042 0.08 119,121,124,177,179,194,198

131492 5551 5.29 0.165 1.04 38031-48117 52 58,62,64,73,77,122,179,185

0.17 0.069 0.06 198

142983 5941 4.88 0.334 0.97 38031-49593 595 2,6,26,36,58,60,62,64,73,

48 Lib -shell 0.07 0.063 0.02 77,110,114,115,139,140,145 170,177,181,182,196,198

148184 6118 4.30 -0.011 1.67 35400-48123 175 2,6,26,58,62,64,77,83,92,

0.07 0.020 0.06 110,114,115,142,170,196,198

162428 B7IV-Ve 0.345 7.12 0.298 0.99 43304-47744 9 2,122,169,198

0.02 0.052 0.03

162732 6664 6.80 0.341 0.89 37666-49234 197 2,5,9,19,33,39,47,49,84,85,

88 Her 0.07 0.029 0.04 89,91,96,110,149,151,160, 170,198

168797 6873 B4IVe 0.220 6.13 0.191 1.06 35400-48328 43 4,28,38,54,110,122,142,160,

0.03 0.011 0.02 166,198

171780 6984 B5Vne 0.264 6.10 0.248 0.90 38913-48375 11 38,110,142,169,170,178,198

0.03 0.013 0.02

173219 B0Ve 0.07: 7.85 0.004 1.49 34271-49594 343 88,94,122,139,140,142,181,

0.02 0.005 0.02 182,198

178175 7249 B2Ve 0.162 5.54 0.175 1.00 35400-48341 36 24,37,58,59,60,62,64,73,

V4024 Sgr 0.05 0.015 0.02 110,122,142,198

183656 7415 B6Ve-shell 0.315 6.07 0.378 1.12 39902-49591 162 19,28,73,91,139,140,181,

V923 Aql 0.03 0.031 0.02 182,198

184279 B0Ve 0.06: 7.01 0.092 1.23 34013-49591 340 2,29,30,40,91,110,113,122,

V1294 Aql 0.11 0.031 0.04 138,140,142,148,161,162, 170,181,181,186,188,198

187811 7565 B2.5Ve 0.170 4.89 0.208 0.82 35400-46664 79 38,54,55,110,114,115,130,

12 Vul 0.04 0.012 0.02 142,146,160,198

191610 7708 B3IVe 0.190 4.93 0.144 0.85 35400-48441 192 1,19,38,54,71,114,115,130,

28 Cyg 0.03 0.012 0.03 142,146,156,160,198

195325 7836 B9e-Shell 0.44: 6.03 0.485 1.08 39902-48372 11 19,91,110,153,171,198

1 Del 0.04 0.038 0.02

200120 8047 4.80 0.052 0.95 32833-49719 660 2,10,15,38,46,50,52,107,

59 Cyg 0.08 0.016 0.06 110,114,115,130,133,134, 142,155,160,167,196,198

205637 8260 4.54 0.203 0.79 35400-49593 449 22,26,28,30,58,59,60,64,73

-shell 0.06 0.026 0.02 78,91,114,115,123,139,140, 142,150,181,182,198

210129 8438 B6-7Vne 0.317 5.71 0.283 0.96 39316-43555 13 19,33,39,110,114,152,165,

25 Peg 0.06 0.031 0.02 169,198

217050 8731 5.31 0.231 0.96 35400-43796 419 2,38,80,85,91,110,142,198

EW Lac 0.05 0.041 0.04

217543 8758 6.53 0.183 0.90 34348-43373 51 38,85,86,91,110,142,198

0.02 0.013 0.03

218674 B3IVeShell 0.186 6.76 0.212 1.09 34348-47028 198 85,86,91,110,160,198

KY And 0.02 0.011 0.02

 

1 Adelman (1992) 67 Ferrari-Toniolo et al. (1978) 133 Lutz & Lutz (1972)

2 Alvarez & Schuster (1982) 68 Franco (1989) 134 Lutz & Lutz (1977)

3 Appenzeller (1966) 69 Frohlich & Nevo (1974) 135 Lynas-Gray & Hill (1979)

4 Ardeberg & Wramdemark (1970) 70 Garrison & Kormendy (1976) 136 Lynga (1959)

5 Baldinelli et al. (1981) 71 Golay (1958) 137 McNamara (1976)

6 Balona et al. (1992) 72 Gray & Olsen (1991) 138 Malmquist et al. (1960)

7 Barbier (1947) 73 Grønbech & Olsen (1976) 139 Manfroid et al. (1991)

8 Barrera et al. (1991) 74 Guetter (1974) 140 Manfroid et al. (1995)

9 Barylak & Doazan (1986) 75 Gulliver A.F. (1983) 141 Margon et al. (1976)

10 Belyakina & Chugainov (1960) 76 Guo et al. (1995) 142 Mendoza (1958)

11 Böhme (1984) 77 Gutierrez-Moreno & Moreno (1968) 143 Menzies et al. (1990)

12 Böhme (1985) 78 Gutierrez-Moreno et al. (1966) 144 Mitchell (1960)

13 Böhme (1986) 79 Häggkvist (1971) 145 Moffet & Barnes (1979a)

14 Böhme (1988) 80 Häggkvist & Oja (1966) 146 Moffet & Barnes (1979b)

15 Bouigue (1959) 81 Häggkvist & Oja (1969) 147 Mook et al. (1974)

16 Božič et al. (1995) 82 Häggkvist & Oja (1970) 148 Moreno (1971)

17 Brodskaja (1968) 83 Hardie & Crawford (1961) 149 Nakagiri & Hirata (1979)

18 Brucato & Kristian (1972) 84 Harmanec et al. (1978) 150 Oblak & Chareton (1980)

19 Cameron (1966) 85 Harmanec et al. (1980) 151 Oja (1991)

20 Catalano & Umana (1987) 86 Harris (1955) 152 Osawa & Hata (1960)

21 Chochol et al. (1985) 87 Harris (1956) 153 Osawa & Hata (1962)

22 Clariá & Escosteguy (1981) 88 Haug (1970) 154 Papousek (1979)

23 Corben (1966) 89 Haupt (1974) 155 Pavlovski et al. (1996)

24 Corben (1971) 90 Haupt & Moffat (1973) 156 Pavlovski & Ružič (1990)

25 Cousins (1962) 91 Haupt & Schroll (1974) 157 Penprase (1992)

26 Cousins & Stoy (1963) 92 Heck & Manfroid (1980) 158 Penston (1973)

27 Cousins (1964) 93 Hill (1970) 159 Percy (1986)

28 Cousins (1965) 94 Hiltner (1956) 160 Percy et al. (1988)

29 Cousins (1973) 95 Hiltner & Johnson (1956) 161 Pfleiderer et al. (1966)

30 Cousins & Stoy (1962) 96 Hirata (1995) 162 Philip & Philip (1973)

31 Cousins & Stoy (1970) 97 Hoag et al. (1961) 163 Poretti (1982)

32 Crawford (1963a) 98 Hogg (1958) 164 Roche et al. (1993)

33 Crawford (1963b) 99 Hopp & Witzigmann (1980) 165 Roman (1955)

34 Crawford & Barnes (1970) 100 Hopp et al. (1982) 166 Rucinski (1987)

35 Crawford et al. (1966) 101 Horaguchi et al. (1994) 167 Salukvadze & Javakhishvili (1995)

36 Crawford et al. (1970) 102 Horn et al. (1996) 168 Schneider (1987)

37 Crawford et al. (1971a) 103 Horn et al. (1982) 169 Schuster & Alvarez (1983)

38 Crawford et al. (1971b) 104 Howarth (1979) 170 Schuster & Guichard (1984)

39 Crawford et al. (1973) 105 Hutchings (1977) 171 Searle (1958)

40 Dahn & Gueter (1973) 106 Hutchison (1974) 172 Sharov & Lyutyi (1988)

41 Danks & Houziaux (1978) 107 Iliev et al. (1991) 173 Sharov & Lyutyi (1992)

42 Dapergolas et al. (1981) 108 Iriarte (1965) 174 Sharpless (1952)

43 de Loore et al. (1979) 109 Iriarte et al. (1965) 175 Shobbrook (1984)

44 Denoyelle (1977) 110 Jaschek et al. (1980) 176 Simonson (1968)

45 Deutschman et al. (1976) 111 Jeong & Lee (1988) 177 Slawson et al. (1992)

46 Divan & Zorec (1982b) 112 Johnson (1964) 178 Sowell & Wilson (1993)

47 Divan & Zorec (1982c) 113 Johnson & Harris (1954) 179 Stagg (1987)

48 Divan et al. (1983) 114 Johnson et al. (1966) 180 Štefl et al. (1995)

49 Doazan et al. (1982a) 115 Johnson et al. (1967) 181 Sterken et al. (1993)

50 Echevarría et al. (1979) 116 Johnson & Morgan (1953) 182 Sterken et al. (1995)

51 Eggen (1955) 117 Juza et al. (1994) 183 Stokes (1972)

52 Eggen (1963a) 118 Kalv (1977) 184 Stoy (1963)

53 Eggen (1963b) 119 Kiehling (1984) 185 Stoy (1968)

54 Eggen (1968) 120 Kilkenny et al. (1985) 186 Tempesti & Patriarca (1976)

55 Elliot (1974) 121 Kozok (1984) 187 Turner (1976)

56 Erro (1969) 122 Kozok (1985) 188 van der Wal et al. (1972)

57 Feinstein (1967) 123 Lake (1965) 189 Vogt (1976)

58 Feinstein (1968a) 124 Landolt (1969) 190 Warman & Echevarría (1977)

59 Feinstein (1968b) 125 Landolt (1971) 191 Warren & Hesser (1978)

60 Feinstein (1970) 126 Lee (1968) 192 Westerlund (1963)

61 Feinstein (1974) 127 Lenouvel & Daguillon (1956) 193 Westin (1982)

62 Feinstein (1975) 128 Lesh (1968) 194 Wiegandt (1984)

63 Feinstein (1980) 129 Lindroos (1983) 195 Zelwanowa & Schoneich (1971)

64 Feinstein & Marraco (1979) 130 Ljunggren & Oja (1964) 196 Zorec (1986)

65 Fernie (1991) 131 Loden (1969) 197 Zorec (1994)

66 Ferrari-Toniolo et al. (1977) 132 Lucke (1974) 198 Geneva Obs. photom. archives

 

2.4. The stellar sample

From all known Be stars (Jaschek & Egret 1982) we have chosen the brightest ones because they were liable to yield more frequent and detailed observations. Among them we preferred those known for their strong spectroscopic variations (Hubert-Delplace & Hubert 1979) or because they had rather strong photometric changes (Feinstein 1975; Feinstein & Marraco 1979; Alvarez & Schuster 1981). We discarded those objects in binaries where variations of Be characteristics are strongly correlated with the orbital phase, such as HR 2142. The studied Be stars are presented in Table 1. The spectral classification is preferentially from the BCD system and in such cases it is indicated with an upper index ^"a". Otherwise, it is taken from the Hipparcos Input Catalogue (Turon et al. 1992); Buscombe (1977, 1980, 1981, 1988); Hubert-Delplace & Hubert (1979); Slettebak (1982), the Bright Stars Catalogue (Hoffleit & Jaschek 1982) and the Supplement to the Bright Stars Catalogue (Hoffleit et al 1983). For each Be star are given: the stellar BD, D_* (an upper index ^"a" indicates BCD values of D_*, otherwise D_* is a mean value for the adopted MK spectral classification); the mean values of V, D and calculated from all collected data and the respective standard deviations (); the JD-2400000 of the first and the last observation; the number of individual triads (); the references of collected data. The standard deviation of variations in the (B-V) and (U-B) colour indices can straightforwardly be derived using the transformations: and .

2.5. Reduction to spectrophotometric parameters

2.5.1. Comparison stars

We took as comparison stars a set of 82 well-observed O, B and A0 type stars in the BCD system, all presented in Zorec & Briot (1991), whose and D parameters are determined to closer than m and dex respectively. We considered stars of luminosity classes from dwarfs to bright giants and with different rotational velocities to get a first and direct insight on the possible scatter that differences in intensities of spectral lines can introduce in the transformations (photometric indices) (spectrophotometric parameters). A large number of these stars (75%) are used either as MK, photometric or spectrophotometric standard/comparison stars (Divan 1966; Golay 1973; Rufener 1979; Hauck 1985; Tüg 1980; Thé et al. 1986; Perry et al. 1987; Zorec et al. 1983; Harmanec et al. 1994). Table 2 (click here) gives the adopted comparison stars, with the and D parameters and the corresponding MK spectral type as derived using the BCD system.

2.5.2. Transformation of photometric data into the and D spectrophotometric parameters

 

HD HR Sp.Type D Notes HD HR Sp.Type D Notes

dex dex

3360 153 B2IV 0.146 0.760 3,4,7 40589 2111 B9Iab 0.266 1.580

4142 189 B5V 0.272 0.760 2,7 41753 2159 B3-4V 0.217 0.790 7

11241 533 B2V 0.144 0.840 1,7 43112 2222 B1IV 0.096 0.700 1,7

16582 779 B2III 0.136 0.760 3 43384 2240 B4Ia 0.107 1.940

19356 936 B7-8V 0.350 0.930 44743 2294 B1II-III 0.089 0.729 4

20365 987 B4V 0.234 1.010 45563 2347 B9V 0.422 0.839

20418 989 B4IV 0.244 0.970 7 46075 2374 B6IV-V 0.316 0.920

21071 1029 B5V 0.270 0.980 7 47240 2432 B1Iab 0.070 1.400 6

21278 1034 B4-5V 0.249 0.980 2,4,7 47432 2442 O8Ia 0.021 1.390 1,6

21428 1044 B4V 0.235 0.940 7 47756 2454 B5-6IV 0.292 0.920

21856 1074 B1IV-V 0.094 1.040 2,7 47964 2461 B7III 0.340 1.030

22928 1122 B5-6III 0.286 0.880 3,7 48434 2479 O9.5Ib 0.042 1.110 7

22951 1123 B0.5V 0.089 1.090 5 48977 2494 B3V 0.196 0.800

23288 1140 B7V 0.340 1.080 4,5 49567 2517 B4II-III 0.195 0.870

23324 1144 B7V 0.342 1.060 2,4,7 71155 3314 A1V 0.487 0.808 2,3,4,5,7

23338 1145 B6IV 0.297 0.970 3,4 74280 3454 B3V 0.181 0.730 3,4,5,7

23408 1149 B6-7III 0.315 1.030 83754 3849 B5V 0.250 0.870 3,4,7

23432 1151 B8IV-V 0.376 1.060 4 87901 3982 B7IV 0.348 0.762 3,4,5,6,7

23753 1172 B7V 0.352 1.070 4,7 100600 4456 B3-4V 0.212 0.820 4

23850 1178 B7III 0.340 0.980 3,5 120315 5191 B4V 0.222 0.750 4,5,7

23923 1183 B9V 0.413 0.960 4 147394 6092 B5IV-V 0.265 0.850 2,4,7

24131 1191 B1V 0.100 1.150 160762 6588 B3IV 0.204 0.750 4,5,7

24640 1215 B2V 0.145 1.010 166182 6787 B2.5III 0.156 0.830 2,7

24760 1220 B0III 0.075 0.760 3,5,7 184915 7446 B1II 0.076 1.040 4,6,7

25638 1260 B0III 0.059 1.940 191692 7710 A0IV 0.502 0.820 6

30211 1520 B5IV 0.251 0.790 7 202850 8143 B9Ia 0.240 0.981 4,7

32630 1641 B4V 0.230 0.780 1,3,4,7 204172 8209 B0Ib 0.048 1.010 7

35468 1790 B2III 0.132 0.710 1,3,7 205021 8238 B1III-IV 0.095 0.710

35600 1804 B9Ib 0.363 1.490 206165 8279 B2.5Ib 0.108 1.620 4,7

36371 1843 B4Ia 0.110 1.790 208218 B2II-III 0.110 1.520 2

36959 1886 B1V 0.115 0.730 208440 B1IV 0.091 1.210 2

36960 1887 B0IV 0.072 0.720 1,7 209339 8399 O9IV 0.052 1.210 7

37016 1891 B3-4V 0.196 0.840 209961 8427 B2III-IV 0.156 0.835

37040 1898 B3V 0.184 0.860 212978 8553 B2III 0.150 0.850 5

37043 1899 O7III-IV 0.035 0.700 3,6 214240 8606 B4-5III 0.252 0.848 2

37209 1911 B1V 0.109 0.760 7 214680 8622 O8:V 0.048 0.790 1,2,4,5

37468 1931 O9V 0.057 0.700 6 217101 8733 B2V 0.132 0.780 2,7

37481 1933 B1V 0.116 0.750 7 218376 8797 B0.5II 0.074 1.050 1,7

37744 1950 B1.5V 0.124 0.790 7 218407 8800 B2V 0.165 0.845 2

38666 1996 O8-9V 0.050 0.700 6 222173 8965 B8IV 0.380 0.833 2,3,4,5,7

 

i) The UBV system.

The best correlation between the UBV colour indices and the BD given in the BCD system was found using Johnson's (1958) Q index, which leads, as in the BCD system, to BD free from interstellar medium (ISM) absorption. We used the definition of the Q index given by Gutierrez-Moreno (1975):

UBV data of comparison stars were taken from Mermilliod & Mermilliod's (1994) compilation. The relation giving the BD from Q_UBV is then:

Figure 1 (click here)a shows D_BCD versus Q_UBV. Relation (2) has a linear correlation coefficient r = 0.992 and leads to a mean (O-C) dispersion dex. The gradient is obtained using the (B-V) colour index as:

The correlation between and is given by:

with r = 0.975 and . Figure 1 (click here)b shows the correlation between the gradient and the (B-V) index.

ii) The UBVRI system.

The BD is obtained from relation (2) as for data in the UBV system. The gradient is now defined by also taking into account the (V-R) colour index, in order to better respect the wavelength interval over which the original gradient is defined. However, with the inclusion of the (V-R) index we cannot avoid introducing a new source of error. The resulting relation is:

The correlation between and is:

with r = 0.968 and . Gradients against are shown in Figure 1 (click here). The UBVRI photometric data used to derive relation (5) are from the "Photoelectric Photometric Catalogue in the Johnson UBVRI System'' (Lanz 1986).

iii) The Strömgren uvby system.

As for the UBV system, we derive the value of the BD by defining an ISM reddening-free Johnson's like index Q in the Strömgren uvby system. This index was determined using the mean ISM absorption law of Mathis (1990) to determine the colour excess factor E(u-v)/E(b-y):

The BD is then given by:

with r = 0.997 and dex. The BD could also be derived using the c₁ index of the uvby system by:

but r = 0.986 and dex. In our work we used relation (8). The gradient is obtained from:

The correlation between and is:

where r = 0.986 and . In Fig. 3 (click here)a we show the relation between D _uvby and Q _uvby, and in Fig. 3 (click here)b the relation between and . Relations (8) to (10) were derived using data given in the Photoelectric Photometric Catalogue (distributed by the CDS) (Hauck & Mermilliod 1975; Grø nbech & Olsen 1976, 1977).

iv) Thirteen-colour photometry.

As in preceding cases, we also defined an ISM reddening-free index using the (m₃₅-m₄₀) and (m₄₅-m₅₂) colour indices, where m₃₅, m₄₀, m₄₅ and m₅₂ are magnitudes respectively at 0.3536, 0.4030, 0.4571 and 0.5183 effective wavelengths of the 13-colour medium-narrow-band photometric system (Johnson & Mitchell 1975). The index is defined as:

The resulting relation for the BD is:

where r = 0.994 and dex. For the gradient the relation is:

The relation between and is given by:

for which r = 0.977 and . In the 13-colour photometric system the V magnitude was obtained as:

whose comparison with the original V magnitude of the UBV system has a correlation coefficient r = 0.999 and mag. Figure 4 (click here)a shows the relation between and Q _uvby, Fig. 4 (click here)b that between and and Fig. 4 (click here)c that between V and .

v) The Geneva photometric system.

The ISM reddening-free Q index corresponding to this system was defined as:

from which the respective BD can be derived as:

where r = 0.985 and dex. It can also be shown that:

where d is a measure of the BD defined in the Geneva photometric system (Rufener 1981); here we have r = 0.978 and dex. In our work we used the value given by (18). The gradient is defined as:

which compares with as:

so that r = 0.964 and . Data for comparison stars concerning the Geneva photometric system were taken from Rufener (1980, 1988). Figure 5 (click here)a shows the relation between and and Figure 5 (click here)b that between and .

2.6. Uncertainties affecting the and D parameters

BDs of Be stars in the BCD system are given with typical errors dex. The width of a D_* interval of a given spectral type-box in the BCD system depends on the effective temperature and on the luminosity class. The mean value of these widths is dex (Chalonge & Divan 1973). Comparing the BCD spectral type determinations of program stars with those given in the MK system for the same objects (Slettebak 1982) we derived the mean absolute spectral type deviation between BCD and MK determinations (Sp.T.) (of sub-spectral type). Hence, we can argue that for those Be stars where only MK classification exists, the value of D_* adopted in Table 1 may be in error by dex.

Among the most important uncertainties affecting the indirect and D parameters derived from relations given in Sect 2.5.2 are: (1) random uncertainties of original photometric data; (2) systematic differences related to the effective wavelengths of photometric magnitudes; (3) presence of variable spectral lines in the filter wavelength pass-bands; (4) extrapolations of D values in phases of strong emission in the second BD component.

(1) - Uncertainties related to photometric measurements are difficult to determine, as it is not easy to compare measurements given in different systems or even in the various versions of the same photometric system (Sterken & Manfroid 1992). Using the mean errors of individual measurements of colour indices in different photometric systems (CETAMA 1978; Crawford & Barnes 1970; Johnson et al. 1967; Manfroid et al. 1991; Rufener 1981; Shuster 1976; Sterken & Manfroid 1992) and the relations between D, and the concerned colour indices given in Sect. 2.5.2, we derived the expected mean errors of individual D and for each photometric system used in this work. They are given in Table 3.

 

BCD 0.035 0.015

Genève 0.040 0.020

uvby 0.070 0.020

13-c 0.085 0.024

UBV 0.054 0.026

UBVRI 0.075 0.026

Table 3: Mean errors of individual and D

Whenever possible, we checked that the photometric indices of B0-A1 comparison and/or check stars in the various photometric systems used or versions of same systems did not introduce systematic scatterings higher than those found when establishing the transformation relations of Sect. 2.5.2. It is however known, mainly for the (U-B) colour indices, that according to different authors high systematic deviations may exist. When such deviations were detected, the data concerned were discarded. Systematic differences are sometimes found among uvby photometric indices if the filters used are not the same. By comparing non variable stars observed in these slightly different systems and/or looking for simultaneous observations of the studied Be stars, we reduced such observations to a common frame. When these reductions were not possible we did no use the existent data.

(2) - The strongest emission in the Balmer continuum appears at m. As the filter-detector response in near-UV is frequently only a few per cent at this wavelength, the reddening effect induced by the CE continuum emission near the BD (scaled by calibrations based on normal B stars of Sect. 2.5) can then be underestimated. Hence, relations like (8) or (9) of the uvby system and (13) in the Thirteen-colour photometry may in principle give slightly underestimated values of the total BD.

(3) - Variations of the (, D) parameters cannot always be attributed to variations of the continuum spectrum alone. Spurious reddening and bluening effects can be produced by variable CE emission/absorption components in lines entering the filter wavelength pass-bands. A very rough upper limit of the reddening effect produced by absorption lines in the (, D) parameters can be estimated by comparing the UBV colour indices of low and high line-blanketed models of stellar atmospheres (Kurucz et al. 1974; Kurucz 1992). Thus, the averaged systematic reddening effect found in the (B-V) and (U-B) indices for log g = 2.0 to 4.5 and for all effective temperatures from to 30000 K leads to a change . Similarly, the highest increase of the BD, dex, is for K. It is however much lower for other effective temperatures, especially among the hottest ones. In actual Be stars, we may then expect that, depending on the relative degree of the line emission/absorption variability produced by the CE entering the photometric filter pass-bands, the (, D) will show reddening or blueing components which do not directly reflect variations of the continuum spectrum only.

(4) - As relations between spectrophotometric D and the photometric colour indices were obtained using stars without emission, when or D < 0 they are necessarily extrapolated. We compared such extrapolated values of D with those obtained in the BCD for similar strong emission phases of HD 5394, HD 24534, HD 60848, HD 148184 and HD 200120 which can be seen in Fig. 6. However, the nice agreement between both determinations do not indicate the existence of any strong deviation.

The total amount of the quoted uncertainties is almost impossible to estimate, because they depend on the star and on its variation phase. However, using as reference the relations derived from BCD system measurements only [which are independent of uncertainties (1) to (4)] of strongly variable Be stars, we can obtain a rough estimate of the mean total amount of possible deviations which affect the indirect and D parameters derived from multicolour filter photometry. Comparing the mean deviations of data in each photometric system with the mean regression lines derived with genuine BCD parameters, we obtain dex and m.

         

Figure 1: a) Transformation of Q_UBV to D_BCD. b) Transformation of the (B-V) colour index to the gradient

 

Figure 2: Comparison of with gradients

 

Figure 3: a) Transformation of to D_BCD. b) Comparison of with gradients

 

Figure 4: a) Transformation of to D_BCD. b) Comparison of with gradients

Figure 5: a) Transformation of to D_BCD. b) Comparison of with gradients